Micro-FEM models based on micro-CT reconstructions for the in vitro characterization of the elastic properties of trabecular bone tissue.

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FACOLT `A DI INGEGNERIA

Corso di Laurea Magistrale in Ingegneria Meccanica Dipartimento di Ingegneria Industriale

Micro-FEM models based on micro-CT

reconstructions for the in vitro characterization of

the elastic properties of trabecular bone tissue.

Sviluppo di modelli micro-FEM

derivati da ricostruzioni micro-CT

per la caratterizzazione in vitro

delle propriet`

a elastiche

del tessuto osseo trabecolare.

Tesi di Laurea in Laboratorio Di Meccanica Dei Tessuti Biologici M

Candidato:

Gianluca Iori

Relatore:

Chiar.mo Prof.

Luca Cristofolini

Correlatore:

Ing. Martino Pani

Anno Accademico 2011/12 Sessione III

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Summary

This master’s thesis describes the research done at the Medical Technology

Laboratory (LTM) of the Rizzoli Orthopedic Institute (IOR, Bologna, Italy),

which focused on the characterization of the elastic properties of the trabecular bone tissue, starting from october 2012 to present.

The approach uses computed microtomography to characterize the architecture of trabecular bone specimens. With the information obtained from the scan-ner, specimen-specific models of trabecular bone are generated for the solution with the Finite Element Method (FEM). Along with the FEM modelling, me-chanical tests are performed over the same reconstructed bone portions. From the linear-elastic stage of mechanical tests presented by experimental results, it is possible to estimate the mechanical properties of the trabecular bone tissue. After a brief introduction on the biomechanics of the trabecular bone (chapter 1) and on the characterization of the mechanics of its tissue using FEM models (chapter 2), the reliability analysis of an experimental procedure is explained (chapter 3), based on the high-scalable numerical solver ParFE. In chapter 4, the sensitivity analyses on two different parameters for micro-FEM model’s reconstruction are presented.

Once the reliability of the modeling strategy has been shown, a recent layout for experimental test, developed in LTM, is presented (chapter 5). Moreover, the results of the application of the new layout are discussed, with a stress on the difficulties connected to it and observed during the tests. Finally, a proto-type experimental layout for the measure of deformations in trabecular bone specimens is presented (chapter 6). This procedure is based on the Digital Image Correlation method and is currently sunder development in LTM.

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Sommario

La presente tesi magistrale descrive l’attivit`a di ricerca svolta presso il Lab-oratorio di Tecnologia Medica (LTM) dell’Istituto Ortopedico Rizzoli (IOR, Bologna, Italia) dall’ottobre 2012 ad oggi, nell’ambito della caratterizzazione delle propriet`a elastiche del tessuto osseo trabecolare.

L’approccio utilizzato è quello della modellazione, con il Metodo degli Elementi Finiti (FEM), di porzioni di osso trabecolare la cui architettura viene ricostru-ita grazie ad una scansione con microtomografia computerizzata. Alla model-lazione FEM è abbinata l’esecuzione di test meccanici sulle stesse porzioni di osso ricostruite. In virtù del comportamento elastico lineare del campione che si evidenzia nella prima fase di caricamento meccanico, è possibile ricavare una stima delle proprietà elastiche del tessuto osseo trabecolare, altrimenti difficil-mente analizzabili per vie sperimentali.

Dopo una breve introduzione sulla biomeccanica dell’osso trabecolare (capi-tolo 1), e sulla caratterizzazione delle sue proprietà meccaniche di tessuto per mezzo di modelli FEM (capitolo 2), nel capitolo 3 si affronta l’analisi sulla affidabilità di una procedura sperimentale basata sull’utilizzo del solutore nu-merico ad alta scalabilità ParFE. Sempre nell’ottica di una conferma della accuratezza delle stime di modulo di tessuto ottenute grazie alla strategia in questione, il capitolo 4 presenta due differenti studi di sensitività su parametri chiave della ricostruzione di modelli micro-FEM di osso trabecolare.

Una volta attestata l’affidabilit`a di suddetto approccio modellistico, viene de-scritto, nel capitolo 5, un layout per test sperimentali sviluppato di recente presso l’LTM. In questa sezione, si affrontano sia i risultati della applicazione del nuovo layout che le criticit`a ad esso connesse e venute alla luce con la con-duzione dei primi test.

In conclusione, nel capitolo 6 viene illustrata una strategia per la misura delle iii

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deformazioni in provini di osso trabecolare fondata sull’utilizzo di un sistema di correlazione di immagini digitali. Tale sistema si trova, alla data odierna, in fase prototipale presso l’LTM dello IOR.

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List of Figures

1.1 Bone Morphology . . . 3 1.2 Cortical Bone . . . 4 1.3 Cancellous Bone . . . 5 1.4 Osteon . . . 6 1.5 Emiosteon . . . 7

1.6 Bone biopsy and histological examination . . . 9

1.7 Histological section . . . 10

1.8 Stress-Strain curve from compressive test . . . 14

2.1 Computed Tomography . . . 21

2.2 Micro-CT scan . . . 22

2.3 Cross-section Histogram . . . 23

2.4 Binarization of micro-CT cross-section reconstruction image . . 24

2.5 From pixels to voxels . . . 26

2.6 Segmentation and connectivity test . . . 27

2.7 Hexahedral element . . . 30

2.8 3D cancellous bone model . . . 30

2.9 Extraction of cancellous bone specimen . . . 31

2.10 Experimental layout: picture . . . 33

2.11 Stress-Strain curve from compressive test . . . 34

2.12 Experimental layout: scheme . . . 35

2.13 Specimen model scheme . . . 36

3.1 Experimental layout of the previous procedure . . . 41

3.2 ParFE predictivity: block diagram . . . 45

3.3 Apparent modulus prediction: full database . . . 47 ix

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3.4 Apparent modulus prediction: residual analysis . . . 48

3.5 Apparent modulus prediction: 2-fold cross-validation . . . 50

3.6 Apparent modulus prediction: ParFE and ANSYS . . . 53

3.7 Apparent modulus prediction: micro-FEM model procedures vs experimental data . . . 54

3.8 Leave-one-out cross-validation: ParFE and ANSYS . . . 55

4.1 BV/TV Normal Distribution . . . 66

4.2 Threshold sensitivity test . . . 70

4.3 Effects of the threshold variation . . . 72

4.4 Voxel Size Sensitivity Test . . . 79

5.1 Displacements Variability . . . 85

5.2 Micro-CT field of view . . . 86

5.3 Experimental layout - photo . . . 87

5.4 Experimental layout - Angular reference markers . . . 88

5.5 Experimental layout - scheme . . . 89

5.6 Experimental layout - postprocessing scheme . . . 90

5.7 Alignment between micro-CT stacks . . . 93

5.8 Apparent modulus from compression tests . . . 99

5.9 Apparent modulus from micro-FEM analysis . . . 101

5.10 Tissue modulus estimations from micro-FEM analysis . . . 102

5.11 Apparent modulus prediction from micro-FEM analysis . . . 104

5.12 Sensitivity study: surface plot . . . 105

5.14 Trabecular discontinuity of specimen 3 . . . 107

5.13 Sensitivity study: 2D plots . . . 114

5.15 Local displacements: specimen 1 . . . 115

5.16 Local displacements: specimen 1, datails . . . 116

5.17 Local displacements: specimen 5 . . . 117

5.18 Local displacements . . . 118

6.1 ARAMIS 5M . . . 120

6.2 Assembly of the DIC based layout . . . 122

6.3 Stochastic pattern application . . . 123

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INDEX xi

6.5 DIC report at frame 0 . . . 127 6.6 DIC acquisition registration procedure: pattern adherence and

surface best-fit . . . 129 6.7 DIC and micro-FEM: comparison of displacements at reference

levels . . . 130 6.8 DIC and micro-FEM: acquisition surface . . . 133 6.9 DIC and micro-FEM: global comparison of displacements . . . . 134

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List of Tables

3.1 Apparent modulus prediction with ParFE: training subset . . . 49

3.2 Apparent modulus prediction with ParFE: testing subset . . . . 49

3.3 FEM prediction: ANSYS and PARFE . . . 52

4.1 Threshold sensitivity: estimated tissue moduli . . . 68

4.2 Threshold sensitivity results: tissue modulus variation . . . 69

4.3 Reconstruction resolution sensitivity test results . . . 78

5.1 Bone structural indexes from 6 specimens . . . 94

5.2 New experimental layout: mechanical tests results . . . 98

5.3 Mean apparent modulus from mechanical tests . . . 100

5.4 Tissue modulus estimations from average displacements . . . 103

6.1 DIC acquisition’s parameters . . . 124

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”Todo hacer es conocer y

todo conocer es hacer”

”Todo lo dicho es dicho por alguien”.

Humberto Maturana R.

Francisco Varela G.

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Chapter 1 Principles of Bone Biomechanics

Bone is a self-repairing and self-remodeling material.

The skeletal system provides support and protection for soft tissues of the body.

It supplies the framework for the bone marrow and allows transmission of forces originated by muscular contraction during the movement. Finally, with its mineral content, bone serves as a reservoir of calcium ions.

Bones, which constitute the skeleton, present various sizes and shapes. De-spite their functional and morphological remarkable differences, all bones are composed by the same elemental structure. Bone tissue, a mineral-base ma-trix characterized by high rigidity and hardness, is the main constituent of the skeletal system.

A sheet of fibrous connective tissue called Periostium covers most of the ex-ternal surface of the bone. Similarly, Endostium, another fibrous connective layer, covers the surface of internal bone cavities.

Cavities are filled with bone marrow, an hematopoietic tissue witch can be distinguished between red and yellow marrow. Finally, the articulating (joint) surfaces at the ends of long bones are covered with a thin layer of articular cartilage, a material presenting extremely low friction coefficient.

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1.1 Bone Composition

Bone is composed of 65% (in weight) mineral and 35% organic matrix, cells and water.

The inorganic mineral fraction is in the form of small crystals with shape of needles, plates, and rods located within and between collagen fibers.It is largely impure hydroxyapatyte mineral Ca10(P O4)6(OH)2, containing carbonate,

cit-rate, fluoride and strontium adsorbed onto the crystal surface.

The organic matrix consists of 90% of collagen and about 10% of various non collagenous proteins.

1.2 Bone Morphology

In long bones the diaphysis, a cylindrical shaft of compact cortical tissue, con-nects two wider ends, called epiphysis, as it can be seen in figure 1.1. The connections between diaphysis and epiphysis are conical regions called meta-physis. In the diaphysis, a central cavity, called marrow cavity, holds yellow marrow. Epiphysis are composed mainly of spongy tissue surrounded by a thin layer of cortical bone. Trabecular cavities are filled with red marrow.

At a microscopical observation, bone tissue of adult mammals reveals to be composed of multiple packages formed of layers of collagen fiber and hydrox-yapatyte. Unit layers are called lamellae.

Typically, lamellae are 3 to 7 µm thick and collagen fibers are disposed ap-proximately parallel to each other. Within adjacent unit lamellar plates, the main fiber orientation can differ by as much as 90◦.

1.2.1 Cortical Bone

Cortical bone is a dense and compact structure that constitutes approximately the 80% of the whole mass of the skeleton system. In long bones, cortical bone composes the diaphysis and the external surfaces of epiphyseal regions. Because of its high stiffness it is the main responsible for the support and protective tasks of the skeleton and for bone’s mechanical strength.

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follow-1.2 Bone Morphology 3

Figure 1.1: Bone morphology: organization and anatomy of cortical and can-cellous bone in human femour is shown. Morphological differences between epiphyseal and diaphyseal regions should be noted.

ings different structural patterns:

 Concentric lamellae forming circular rings, in the osteon.

 Circumferencial lamellae: a thin lamellar layer enveloping bone external surface.

 Interstitial lamellae: angular fragments of concentric lamelale, residuals of previous Haversian systems eroded by bone remodelling.

1.2.2 Cancellous Bone

Cancellous bone is a low-density structure that composes metaphyseal and epi-physeal regions in long bones. In cancellous bone (also called trabecular bone),

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Figure 1.2: Long bone shaft compact bone representation with a detailed view of the Haversian system. Cortical bone forms the outer wall of all bones, being largely responsible for the supportive and protective function of the skeleton. Approximately 80% of the skeletal mass in the adult skeleton is cortical bone.

bone tissue is organized in a complex, high-connected net of small elements called trabeculae. In this structure, cavities are filled with red bone-marrow. Morphology of cancellous bone differs consistently depending on several fac-tors such as subject age, skeleton regions and pathologies. Trabecular mean morphology can vary between a thin, regular, rod-type shape and a plate-like geometry.

The structure has a principal, uniform orientation of trabecular geometry which is aligned with main loading directions of the bone.

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1.2 Bone Morphology 5

Figure 1.3: Coloured scanning electron micrograph of cancellous bone. A percentage of around 20 % of the bone mass is cancellous bone: a lattice-type structure of plates and rods. The structural unit of cancellous bone has typical mean thicknesses ranging between 50 µm and 300 µm and is known as trabecula. Trabecular bone is found in the inner parts of the skeleton.

1.2.3 Bone Structural Units

The main structural unit of cortical bone is the osteon, a cylindrical, hollow column formed of concentric rings of lamellae.

In the osteon (see figures 1.2 and 1.4), the central canal (called Haversian canal) allows blood and lymphatic vessels and nerves to pass.

Haversian canals are connected to each other as like as with marrow and pe-riosteum by transverse Volkmann’s canals.

The typical osteon has an external diameter of 200 to 250 µm with a lamellar circular wall of approximately 70 to 100 µm thick [44]. It’s esternal surface is covered with a 1 to 2 µm thick layer of mineral collagen fibers, called cement line.

In addition, cortical bone shows a dense net of small cavities running lon-gitudinally inside osteon’s walls called lacunae. Lacunae are connected with eachother by transverse canals called canaliculi.

The dense, interconnected net of lacunae and canaliculi, contains a network of bone cells (Osteocites) entrapped as a result of osteon formation.

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Figure 1.4: Detail of compact bone tissue.

It has been suggested that the osteocite network acts as a detector of bone microfracture and can activate, through biochemical signals, the mechanism of bone resorption and remodelling.

In cancellous or trabecular bone the structural unit is the trabecular packet itself, also called emiosteon (see figure 1.5).

The ideal trabecular packet is a 1 mm long and 50 µm thick cilinder-like bar connected with the structure with a radius of about 600 µm.

As with osteons in cortical bone, a cement line covering holds trabecular packets in cancellous bone.

Lamellar tissue of the hemiosteon appears to be aligned parallel with trabec-ular surface and hosts the typical lacunae with osteocites inside.

1.3 Cancellous Bone Mechanics

Bone is a non homogeneous material composed by both organic and inor-ganic substances. Its mechanical behavior is the result of the interaction of its inorganic solid (mineral) phase with the organic phase, which is made of collagen fibers. Trabecular composition is generally similar to the composition of osteons in cortical bone, but with a lower degree of mineralization and

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con-1.3 Cancellous Bone Mechanics 7

Figure 1.5: Polarizing microscopy image of the longitudinal section of an emios-teon. Lamellar fibers with different orientation present different brightness under polarized light. The presence of lacunae hosting osteocite cells can be noted.

sequently lower density (1.874 g/cm3 _{for trabecular tissue versus 1.914 g/cm}3

for cortical tissue). The content of water being slightly higher in cancellous tissue (cancellous = 27 %; cortical = 32 %).

It is no doubt true that the main feature distinguishing cancellous from corti-cal bone is the typicorti-cal trabecular organization in an interconnected and high porosity structure. As consequence of this fact, cortical bone volume fraction ranges between 85 95 %, as opposed to 5 60 % for trabecular bone.

1.3.1 Architecture

In trabecular bone, geometrical as well as physical properties can vary with anatomic site, age, gender, pathologies and more. It is therefore fundamental to possess solid methods for the estimation and the characterization of the trabecular structure. This can be done through the description of the following properties of cancellous bone:

bone density

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trabecular thickness trabecular geometry

The estimation of the above mentioned characteristics of cancellous structure is a point of prior importance for the understanding of bone tissue mechanical properties and of phenomena like bone resorption and remodelling.

Two different approaches exists for the estimation of such parameters: 2D histomorphometric methods

3D reconstruction methods

Let’s have a more detailed view of both methods and relative bone indexes that can be defined with.

Traditional 2D histomorphometric methods

Histomorphometric methods are based on the direct microscopical observation of specimens from hystological sections. This kind of method is considered as “gold-standard” for the direct determinations of structure parameters.

Bone biopsies have to be embedded in a resistant cement (commonly poly-methylmetacrylate, PMMA) to allow 20 µm thick sectioning for microscopic observation (see figure 1.6). Images obtained with gold-standard technique (figure 1.7) can reach a definition of 4 µm per pixel.

After selection of the region of interest (ROI) the gray-scale image undergoes to a binarization process called segmentation where bone pixels are distin-guished from non-bone ones by selecting a gray-scale value as limit level for bone tissue. From the binary image (figure 1.7), the following parameters describing trabecular architecture can be estimated:

 Tissue Area T.Ar [mm2_{] Total area of the ROI.}

 Tissue Volume TV [mm3_{] Total volume of the ROI.}

 Bone Area B.Ar [mm2_{] Total area of bone in 2D cross-section.}

 Bone Volume BV [mm3_{] Total volume of bone in 3D cancellous model.}

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1.3 Cancellous Bone Mechanics 9

Figure 1.6: Bone biopsy and preparation of the histological observation pro-cedure.

 Bone Surface BS [mm2_{] Total bone surface in 3D trabecular model.}

From a stack of images a 3D ROI is assembled and the following primary indexes calculated:

 Bone Volume Fraction

BV /T V = Bone V olume

T otal V olume (1.1)

Bone density, or bone volume fraction, can be calculated equivalently by the area fraction, the line fraction or the point fraction [4]. This means that there is no requirement for the orientation of planar or linear section probes. Typically, trabecular bone BV /T V varies between 10% and 30%.

 Bone Surface Density

BS/T V = Bone T otal Surf ace

T otal V olume (1.2)

BS/TV allows the quantification of bone-marrow interface of trabecular bone. Considering a section of the bone specimen, BS/T V is related to

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Figure 1.7: (a) Histological section of a bone sample containing both cortical and cancellous bone. The dashed line include the region of interest (ROI) (4mm x 4mm in size) containing only cancellous bone. (b) Binary ROI: black pixels identify bone, the white ones identify background.

the linear length of the interface between bone and marrow per section surface.

A set of secondary indexes is useful for the description of trabecular geometry. It is common, in literature [4], to characterize cancellous bone structure by the assumption of two opposite ideal trabecular models. These are the plate-model and the rod-model.

For the calculation of the following secondary indexes, the plate-model as-sumption is used for trabecular bone.

 Trabecular Thickness (µm) The main trabecular thickness with the plate-model structure assumption.

T b.T h = 2 1.199 B.Ar B.P m = 1 2 BV T S (1.3)

 Trabecular Number (or trabecular density) (1/mm) The number of trabecular planes (in a plate-model structure) per unit length, along a direction normal to the plates.

T b.N = 1.199 2 B.P m B.Ar = 1 2 BS T V (1.4)

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 Trabecular Separation (µm) Is the distance within two plates in the plate-model trabecular structure.

T b.Sp = 1

T b.N − T b.T h (1.5)

A condition for the histomorphometric observation, is the analysis of entire bone cross-sections. The observation is usually conduced over a reduced num-ber of sections, due to the difficulties in slice’s extraction. For this reason, the applicability of 2D histomorphometric methods is generally reduced.

3D methods

Three dimensional methods are based on the direct calculation of structure indexes on a 3D reconstruction of the bone specimen. Later, in this chapter, we will se how a 3D model of trabecular bone can be obtained from a micro X-ray Computed Tomography (micro-CT) data set.

Bone indexes can easily be obtained from 3D reconstructions with no use of structure geometrical assumptions. They’re affected by an uncertainty witch is related only to the resolution of the micro-CT scan. We will introduce, here, structure indexes from 3D reconstructions of cancellous bone, refering to section 2.2 for a detailed view of the generation of 3D cancellous bone reconstructions from micro-CT data sets.

 Model Independent Trabecular Thickness (T b.T h∗) (µm) Taken a trabecular structure and a point of it, trabecular thickness can be locally defined as the diameter of the maximum sphere fully included within the structure and including the point itself. A mean value for T b.T h∗ is obtained by averaging over the whole volume.

 Model Independent Trabecular Separation (T b.Sp∗) (µm) Can be calculated with the same technique described for Tb.Th considering points of the marrow cavity.

 Mean Intercept Length (MIL) MIL method quantifies the structure interface anisotropy.

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trabecular structure with orientation ω. The number of interseptions of the bone-marrow interface with the grid (I(ω)) is counted. The mean length separing two interseptions is calculated as:

M IL(ω) = L

I(ω) (1.6)

Where L is the total length of the grid line. It is clear how MIL(ω) varies with the angle ω for structures with a certain rate of anisotropy.

A generalization of this technique for the 3D specimen space leads to the definition of an ellipsoid that can be expressed as the quadratic form of a second-rank tensor M. Cowin defined a MIL fabric tensor H witch as the inverse square root of M [4]. The H eigenvalues coincide with the

MIL values in the main directions and it has been found that the larger

values of H are associated with Young’s Modulus maximum values of the structure.

 Degree of Anisotropy (DA) It can be calculated as the ratio between the length of the major and minor axes of the ellipsoid described by the vector MIL.

DA = M ILmax M ILmin

(1.7) Further seondary indexes can be obtained from 3D reconstructions. These are measures of Trabecular Connectivity and Bone Apparent Density. It is clear how 3D reconstruction allows an insight into trabecular structure and an accurate description of its properties thanks to the introduced indexes. Informations on trabecular bone morphology and anisotropy are prior for the better understanding of bone stress behaviour and mechanical properties, as it will be seen later with more details.

1.3.2 Apparent Mechanical Properties of Cancellous Bone

As we have seen in the previous paragraphs, cancellous bone presents a porous an interconnected structure which can vary largely, in shape and organization, with several subject-specific and zone-specific factors.

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be studied as a continuum media independently of its microscopic mechanical behavior. At this scale, trabecular bone exhibits mean mechanical properties that can be referred to the volume of bone under examination. These proper-ties are commonly known as ”apparent” properproper-ties.

1.3.3 Asymmetry

At the apparent level, trabecular bone exhibits strength asymmetry as well as asymmetry in its failure behavior between compressive and tensile loading. In figure 1.8 the stress-strain curve is plotted for a compressive test of human cancellous bone.

In compression, yield is generally reached around = 0.05. A plateau of deformation with constant stress is present and failure strains can reach the 50 %[4].

For tensile loadings, failure occurs earlier ( < 0.03) and the material presents brittle failure behavior. On the other side, its elastic modulus is substantially the same in tension and compression.

1.3.4 Anisotropy

The mechanical apparent anisotropy of cancellous bone is inherently connected with the highly anisotropic nature of its geometrical organization.

The degree of structural anisotropy of cancellous bone (for which an estimation can be calculated with equation 1.7) can vary with age, bone site and patho-logical factors, between others [4]. Similarly, the elastic modulus of cancellous bone can vary significantly with bone site and, coherently with trabecular or-ganization, with the direction of applied load.

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−0.2 −0.15 −0.1 −0.05 0 −8 −7 −6 −5 −4 −3 −2 −1 0 ε [microstrains] Stress σ [N/mm 2 ]

Figure 1.8: Stress-Strain curve for an experimental failure compressive test. Apparent modulus is obtained from the slope (in red in the graph) of the initial quasilinear loading region of the curve.

1.3.5 Relations between bone volume fraction and

elas-tic properties

From the previous paragraphs it is clear how the elastic properties of cancellous bone vary with its density, geometrical organization and with the composition of its tissue. It has been demonstrated how bone density and trabecular orga-nization (through the coefficients BV/TV and MIL presented in section 1.3.1) can account for between the 85 % and the 97 % of the variability of cancellous bone elastic properties [4]. Both linear and exponential relations have been proposed for the dependency between stiffness and bone volume fraction, the coefficients of the equations varying with bone site.

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1.3.6 Assessment of Cancellous Bone Tissue Elastic

Con-stants

The small dimensions of cancellous bone structural units (trabecular thick-ness = 50 µm), make the investigation of tissue level mechanical properties of this kind of bone an extremely hard task for the researcher. As we presented above, the proportionality between apparent stiffness and bone density have been suggested, thus considering the tissue elastic modulus of cancellous bone homogeneous and generally similar to cortical tissue modulus.

Recently developed experimental techniques, based on the combination of me-chanical tests with FEM modeling, have made it possible to characterize the trabecular bone tissue modulus[19, 38, 10, 34, 13]. On one side, this kind of investigation have led to a wide range (from 0.76 to 20 GPa) of cancellous tissue modulus values reported in the literature [34](3.5 8.6 GPa), [19](5.5 -7.7 GPa), [33, 26, 10, 46]. On the other hand, investigators seem to agree in reporting a cancellous modulus lower than cortical modulus of between 10 % and 20 % [4].

We will present here most common techniques in use for the characterization of trabecular tissue modulus of elasticity.

Uniaxial Tensile and Bending Tests

Several delicate tensile testing systems have been designed for the determina-tion of cancellous tissue modulus. Testing of such small specimens is associated with several technical difficulties like the irregular specimen’s geometry and the alignment of the sample with the external load. For these reasons, tensile and bending tests seem to underestimate trabecular tissue modulus [4].

Ultrasonic Techniques

The origin of these techniques is found in the principle by which the speed at which sound travels through solid matter depends on its elastic properties and density. Ultrasonic technique can be applied either to the continuum level cancellous volume or to a microspecimen to measure the stiffness of cancellous tissue. In both cases, its application provide very similar modulus values for

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cancellous bone tissue, which are significantly lower than the corresponding cortical bone modulus.

Micro-Indentation and Nano-Indentation

Micro-indentation of bone tissue was first introduced in the fifties. During the last 20 years Nano-indentation has been used for the determination of bone tissue properties. With an indenter of 2 µm in size and a depth-sensing resolution of nanometers, the unloaddisplacement curve is used to calculate a material elastic modulus. Roughly speaking, nano-indentation has given the higher values for the bone tissue modulus [4].

Numerical Back-calculation with Finite Element Models

From the early seventies the modeling of bone has been proposed as a tool for the investigation of its mechanical behavior.

With the advances introduced by new micro-imaging technologies such as micro-CT, it became possible to accurately describe the 3D microscopic struc-ture of trabecular bone. At the same time we assisted to the development of powerful supercomputers that are now able to solve FEM models characterized by millions of nodes.

The direct conversion of microscopic images from tomographic techniques into hexahedral element based micro-FEM models is at the date a standard ap-proach for the investigation of trabecular bone tissue properties[38, 37, 13, 39, 41].

The process is based on non-destructive mechanical tests (generally uniax-ial compression) of cubic or cylindrical trabecular bone samples.

The experimental apparent modulus of cancellous bone (Eapp exp) is calculated

from the results of mechanical test.

Assuming a tentative value for the cancellous bone tissue modulus of the µ-FEM model (Et F EM), the predicted apparent modulus Eapp F EM of the model

can be determined by simulation of the same boundary conditions as in com-pressive tests. Within the assumption of linear elasticity, the effective

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cancel-1.3 Cancellous Bone Mechanics 17

lous bone tissue elastic modulus Et ef f can then be determined by:

Et ef f =

Eapp exp

Eapp F EM

Et F EM (1.8)

In this process, errors may be introduced by experimental artifacts such as fastening of the specimen’s endcaps and the measurement of trabecular dis-placements with the use of an extensometer.

Recently, the development of experimental layouts free of this kind of artifacts, has led to the estimation of values of cancellous bone tissue modulus the 10 % lower than the modulus of the cortical component of the corresponding bone section.

In the next chapter, we will introduce the experimental based technique used at Rizzoli’s Orthopedic Institute Laboratory of Medical Technology for the determination of cancellous bone tissue mechanical properties. As we will see, the process in use is based on the reconstruction of micro-FEM voxel models of trabecular samples from the data sets obtained through micro-CT scanning. The estimation for the tissue modulus is obtained by use of the back-calculation procedure introduced here.

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Chapter 2 From Micro-CT data sets to 3D

Micro-FEM models

Trabecular bone distributes load from articular surfaces to cortical bone through its bone matrix. The trabeculae in this matrix constitute the actual load-carrying construction.

In the previous chapter we’ve seen how the material properties of the tra-beculae, in combination with their architecture, determine the strength and stiffness of trabecular bone under loading.

From the late eighties, several attempts have been described in the literature to determine the trabecular tissue material properties. The purposed techniques include traditional tensile and bending tests applied to single trabeculae, four point bending experiments, microtensile and ultrasonic techniques.

A more precise way for the determination of tissue stresses and strains can be achieved if the architecture of a large portion of cancellous bone can be represented in detail through a numerical 3D finite element model.

Computed Tomography and the recent introduction of micro-CT enable the scanning of bone specimens at high-resolution (10-30 µm), which allows de-tailed visualization of the trabecular structure.

By use of a ”pixels to voxels” conversion technique, high-resolution (or micro-) FE models incorporating trabecular architecture can be generated.

The advancement in computer hardware and solving strategies obtained in the 19

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nineties made it is possible to analyze now the elastic behavior of large1 _bone

regions with micro-FEM models [38, 1, 9, 30, 37, 36, 39, 40].

In the present chapter we will follow the steps that lead to the generation of a micro-FEM model of a trabecular bone specimen from its high resolution

micro-CT scanning.

2.1 Computed Tomography

X-ray computed tomography (CT) is a medical imaging technique which allows reproduction of cross-section (tomographic) images of a certain part of the body. The process (invented by nobel prize Sir. Godfrey Hounsfield in 1967) is based on projection data obtained from the attenuation of X-rays radiations. Algorithms allow back-calculation of cross-section images of the object from projection data. With proper calibration of the system, cross-section images can then be converted in density (Hounsfield Units) images.

1_{In recent studies [42], the in-vivo scanning through pQCT (peripheal quantitative}

com-puted tomography) was proposed as the basis for the reconstruction of micro-FEM models (isotropic voxel size = 82 µm) of entire bone regions whole bone segments (generally distal radius). the micro-FEM modeling of the distal radius as a comparison with homogeneous bone FEM models.

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2.2 Imaging of micro-CT data-sets 21

Figure 2.1: Computed Tomography: scheme. Projection data obtained from the attenuation of X-rays radiations is converted in density (Hounsfield Units) images from proper algorithms.

Micro CT was introduced in the late 1980’s. Its process is based on a

compact fan-beam type of tomograph and the full 360°rotation of the mecha-nism around the specimen to be scanned.

Each micro-CT image (figure 2.2) reproduces a 2D distribution of linear atten-uation coefficients dependent on the atomic composition of the scanned ma-terial. In modern devices, this leads, after proper calibration and convertion of attenuation coefficients, to a cross-section density image of extraordinary nominal resolution (pixel size = 4 to 30 µm for gold standard devices). For high resolution micro-CT scanning of cancellous bone, cylindrical or pris-matic specimens with a diameter (or side length, respectively) of the order of 5 to 10 mm should be used.

In this thesis, we will refer to the micro-CT scanning procedure in use at the Medical Technology Laboratory of Rizzoli Orthopaedic Institute.

2.2 Imaging of micro-CT data-sets

The technique in use at Rizzoli’s Institute facilities is based on the scanning of cylindrical cancellous bone specimens (generally: Height = 20mm; Diameter

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= 10mm, [27, 28]) from biopsy of human femural head thank to a Skyscan micro-CT2 _{with 0,90}_{°rotation step over a total acquisition angle of 185°.}

The result of the reconstruction from micro-CT attenuation data is a stack of 1024 x 1024 [pixels] grey-scale slice images (pixel size = 19.5µm), with a z-axis step of 19.5µm. An example of cross-section image is shown in figure 2.2.

0 mm 10.0 mm

10.0 mm

Skyscan_1072

Figure 2.2: Micro-CT scan output: cancellous bone cross-section grey-scale density image. The region of interest (ROI) represents only the central part of the output image.

2.2.1 Binarization of cross-section images

The distribution of grey-scale attenuation coefficients of a typical cross-section micro-CT reconstruction image is shown in figure 2.3. Grey levels are inher-ently related with the density of the media which is scanned. The higher peak of the histogram corresponds to image background and marrow regions, while the lower peak to bone tissue.

It is important, when analyzing the histogram of cancellous bone cross section attenuation coefficients (figure 2.3), to focus on the wide range of grey-scale values that is displayed. This is caused by bone density’s natural

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2.2 Imaging of micro-CT data-sets 23 0 1000 2000 3000 4000 0 50 100 150 200 250

Figure 2.3: Density histogram of the grey-scale attenuation coefficients from the cross-section micro-CT scanning of cancellous bone. The monochromatic grey scale is related with density of the media which i scanned. The lower den-sity peak represent bone tissue while the higher bone marrow and background. ity and the uncertainty related to the micro-CT process of data acquisition and conversion. In reconstruction images (see figure 2.2), bone edges are blurred. This makes not trivial the task of tracing bone-marrow interface. In order to permit 3D reconstruction of cancellous bone structure, every image of the stack must undergo a binarization process called segmentation or thresholding, where bone tissue is distinguished from the non-bone background.

Different thresholding techniques have been proposed in the literature [17, 7, 43]. A distinction is usually made between local and global methods.

 Global threshold Global thresholding techniques assume a unique thresh-oldign gray-scale value for the binarization of the data set. The informa-tion on the thresholding level can derive from external histological ob-servations. Alternatively, the level can be chosen so as to match BV/TV from 3D reconstruction calculation with BV/TV determined from direct

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Figure 2.4: Binarization of cancellous bone cross-section data sets: on the left, the region of interest is shown before its binarization. On the right, the same reconstruction image is shown after the segmentation process. Localization of the bone-marrow interface will depend on the grey value chosen as threshold.

observation through histomorphometric methods (see section 1.3.1).  Local threshold Local methods try to escape the difficulty of global

matching by locally detecting the bone-non bone interface. This can be obtained with an imaging differential analysis for geometry detection. An example of local technique is the iteration of gradient-based geometry detection and connectivity test processes [43].

Kim, Zhang and Mikhail [17], investigated the effects of different threshold-ing techniques on micro-CT based trabecular bone models. Particularly, three thresholding procedures were considered: the global (standard) thresholding technique, a Match-global technique, based on matching of bone volume frac-tion from physical data, and an Adaptive technique, where severeal thresholds are obtained from regional histograms. Their findings suggest that predictions of mechanical structural properties of cancellous bone agree well with experi-mental measurements regardless of the choice of thresholding methods. On the other hand Hara, Tanck, Homminga and Huiskes [7], investigated the influence of global threshold variations on the estimated mechanical and struc-tural properties of cancellous bone. Their results confirm how threshold se-lection is important for the accurate determination of volume fraction and mechanical properties, especially for low bone density.

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2.3 Generation of a 3D micro-FEM model 25

subject-specific cancellous bone structure determining trabecular thickness and, consequently, trabecular connectivity. Separate thresholding methods may lead to different models with substantially different mechanical properties [17]. Similarly, for the global approach, the choice of the gray-scale threshold-ing value may have an impact on the resultthreshold-ing properties of the model [43, 7].

2.3 Generation of a 3D micro-FEM model

The stack of cross-section images obtained from micro-CT scanning is assem-bled, by use of a dedicated software, to generate a 3D gray-scale model of the trabecular structure. After this, the segmentation is realized with a deter-mined global threshold.

The procedure performed in Rizzoli’s Medical Technology Laboratory fol-lows the points that are detailed below.

1. From pixels to voxels The stack of cross-section reconstruction images from micro-CT is imported to a medical imaging manager software. Ev-ery image pixel is converted to a volume voxel, where (cubic) voxel size is equal to the native image pixel size (19.5 µm) (see figure 2.5). Each voxel of the 3D model is assigned with the grey value possessed by the corresponding pixel of the original image.

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Figure 2.5: From pixels to voxels: each pixel of the reconstruction absorption image is converted to a cubic voxel. The height of each voxel corresponds to the distance between each section given by the scanner: 19.5 µm with Skyscan 1072 (Skyscan 1072r, BRUKER, Kontich, Belgium)

2. Volume re-sampling To reduce the total number of nodes of the FEM model together with computational cost, the assembled volume can be re-sampled to lower resolution: from the native voxel size of 19.5 µm to 39.0 µm. In this case, each re-sampled voxel will be assigned with the average grey-scale value of all native voxels included by this one. A sensitivity analysis have been conduced on a random set of specimens to investigate the influence of the re-sampling procedure on the estimation of the tissue elastic properties of cancellous bone (see section 4.2). This was found to be negligible for all the considered cases.

3. Thresholding The segmentation of the 3D volume is performed by use of a global thresholding technique, applied a fixed gray-sacle threshold value of 144 (see figure 2.6). The grey-level for the threshold have been obtained from a robust analysis previously realized at IOR’s laboratories[29]. This was based on a comparison between the reconstructed micro-CT images (after their segmentation) and microscopic pictures of the corre-sponding histological sections.

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marrow cavity ad have to be excluded.

While for CT data, a standard calibration tool [14] is available for the conversion of Hounsfield Units (section 2.1) into mineral density, at the date, such a feature is not accessible for the micro-CT in use at Riz-zoli’s Medical Tecnology Laboratories. For this reason, no information is available on bone density at the microscopic scale and all voxels will be assumed to possess the same material properties.

4. Connectivity test A connectivity test is performed (figure 2.6). Here, only a single, interconnected cluster of voxels is maintained, eliminat-ing non-connected bodies and thus defineliminat-ing a contiuous mono-connected structure model (this is a requirement for the FEM numerical analysis). Usually, the portion of non-connected structures discarded by the con-nectivity test represents between the 0.5% and the 2.5% (in number of elements of the micro-FEM model) of the entire reconstruction.

Figure 2.6: Segmentation and connectivity test for the generation of continu-ous trabecular structure voxel model: a schematic view. The voxel volume (a) is assembled from reconstruction images: a grey-scale value is assigned at each voxel. Segmentation process: all voxels with a grey level lower than the thresh-old level are removed (b), uniform property is assigned to all remaining voxels and the remaining non-connected regions are removed through connectivity test (c). A uniform voxel-based model (d) is obtained.

5. Meshing, assignment of material properties and boundary

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voxel is assumed to be an hexahedron-shaped 8-nodes finite element (see section 2.3.1), usually referred to as brick element. Elements will be cu-bic due to the isotropic nature of voxels obtained from micro-CT dataset reconstructions. The basic model material (bone tissue) will be consid-ered to be isotropic and homogeneous and generally characterized by a linear-elastic behaviour. Therefore, the constitutive law for bone tissue is uniqueley defined by only two parameters: elastic modulus and Pois-son ratio. Tentative value are assumed for the parameters (Et F EM [GPa]

and ν). The first is set at a uniform value of 19 GPa while the latter at 0.3 .

Once geometry and material properties of the hexahedral brick element FEM model are defined, external loading and boundary conditions are applied consistently with the experimental layout:

 Lower boundary Considering the bottom of the model of figure 2.12, displacements of all nodes laying under the PMMA cement height are fully constrained leaving no degree of freedom.

 Upper boundary At the same time, all nodes laying on the upper surface of the cylinder-shape model have horizontal displacement fully constrained. The vertical displacement is imposed, to simulate the uni-axial compressive behaviour of the experimental test

2.3.1 The eight-node hexahedral element

For the micro-FEM analysis of bone, tissue is modeled through the linearized elasticity equations. As it has been shown above, the procedure is based on equally shaped micro finite elements, obtained by simply converting all bone voxels to equally sized 8-node brick elements. The discretization of the elastic-ity equations on this domain by means of piecewise trilinear polynomials leads to a linear algebraic system of the form:

KU = R (2.1)

where K denotes the global stiffness matrix, and is obtained through the as-sembly of the stiffness matrix of each element.

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Figure 2.7 represents the 8-node hexahedral element with the system of natural coordinates (ξ, η, ζ). According to this coordinate system, the ele-ment’s faces are defined by: ξ =±1, η = ±1 and ζ = ±1.

The coordinates x of each point of the hexahedral element e can be expressed through the interpolation of its nodal coordinates X with the coordinate shape functions N_C(e):

x(e) = N_C(e)(ξ, η, ζ)X (2.2) Similarly, the displacement field can be expressed through the interpolation of nodal displacements U with the displacements shape functions N_D(e):

u(e) = N_D(e)(ξ, η, ζ)U (2.3) Invoking the isoparametric concept, it is possible to use the same shape func-tions for the description of both element coordinates and displacements:

N_C(e)(ξ, η, ζ) = N_D(e)(ξ, η, ζ) = N(e)(ξ, η, ζ) (2.4) Being the displacements discretized by means of a trilinear polynomial of the form:

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Figure 2.7: 8-node hexahedral ”brick” element: the system of natural co-ordinates is defined by centroids of opposite faces, laying its origin on the element’s centroid. In the natural system (ξ, η, ζ) each point of the element has coordinates in the range [-1,+1]. Nodes at ζ = −1 are first numbered counterclockwise, followed by nodes at ζ = 1.

Figure 2.8: A 3D cancellous bone cubic voxel model. The colormap represent-ing the reconstructed micro-CT attenuation coefficient.

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2.4 Experimental compressive test 31

2.4 Experimental compressive test

We will consider here the mechanical compressive test procedure performed at the Rizzoli’s Orthopedic Institute facilities for the determination of the appar-ent elastic modulus of each bone specimen [27, 28].

Cylindrical specimens of trabecular bone, with a diameter of 10mm and a height of 20mm, are extracted from donor’s femoral heads by means of a bore diamond-coated milling cutter with the cylinder axis aligned with the Mean Trabecular Direction [28] (see figure 2.9). Bone slices are immersed in water during the milling procedure. After their extraction, specimens have to be left immersed in Ringers solution for 24 hours to ensure the rehydratation of the bone tissue. Specimen free length and diameter are measured. Thereafter, each cylinder is scanned by micro-CT (Skyscan 1072r, BRUKER, Kontich, Belgium) following a validated protocol [28], [27].

Figure 2.9: (Left) scheme of femural head with the inscribed 26mm thick bone slice, having the planes oriented orthogonally to the main trabecular direction (MTD, dashed-dotted line) determined previously. The bone slice is cut out from the head, and then the cylinder is drilled out from the slice. (Right) The cancellous bone cylinder extracted (diameter 10mm, height 26mm).

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Before testing, specimens are immersed in Ringers solution for an addi-tional hour. Bone specimens are cemented directly onto the testing machine (Mod. 8502, Instron Corp., Canton, MA, USA) to ensure the alignment be-tween the loading direction and the specimen axis.

Average strain is measured with an extensometer (Mod. 2620-601, Instron Corp., Canton, MA, USA) attached with two rubber bands directly to the central part of the cylinder. The experimental layout for compressive test and the assembly of the extensometer are shown in figure 2.10. It is important, for the compressive test of cancellous bone specimens, to minimize end-artifacts and side-artifacts which can influence the experimental results [35], [3], [15], [20]. The described technique allows the accurate determination of apparent strain since its measurement is unaffected by end effects [15]. Strain rate3 _is

controlled during the test and maintained at 0.01s−1 [28, 20, 40, 22].

The mean elastic modulus of the specimen, usually referred to as apparent elastic modulus, is calculated as the mean slope of the loading curve in its linear elastic region.

In a late version of the experimental protocol (see chapter 5), the test is re-peated four times, along four different angolar positions of the specimen. A system of aluminium markers embedded into the PMMA cement allows the identification of 4 angolar directions on the cylinder-shaped surface of the spec-imen separated by 90°each. The final elastic modulus will then be extimated as the mean value of the modulus registered for each angolar positioning. Subsequently, a failure compressive test is performed, to gain further informa-tions on the yielding curve and post-elastic behaviour of the specimen.

In figure 2.11 a plot of the stress-strain curve from experimental failure com-pressive test of a cylindrical cancellous bone specimen is shown. A first order regression line (plotted in red in the figure) is fitted to the quasilinear region to observe a correlation coefficient R > 0.99. The apparent-level elasticity modulus of the bone specimen (Eapp exp [MPa]) is calculated as the slope of

this regression line.

3_{Strain rate represents the speed of the load application during mechanical tests. The}

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2.5 Solution of FEM model 33

Figure 2.10: Experimental layout for compressive test of a cylindrical cancel-lous bone specimen (diameter = 10 mm; free length = 20 mm). Specimen’s endcaps are embedded into PMMA cement. Displacements at two levels of the specimen are registered with an extensometer (Mod. 2620-601, Instron Corp., Canton, MA, USA). The contact between cylinder and extensometer knives is ensured thank to two rubber bands.

2.5 Solution of FEM model

The aim of the micro-FEM model presented in section 2.3 is that of describing the mechanical test of figure 2.10.

Thanks to the linearity of the FEM model, it is possible to determine the effective tissue modulus by scaling the numerical results with those obtained from experimental testing [4, 40, 38, 1].

Displacements are imposed to the upper surface of the model, replicating the loading conditions of experimental compressive tests. The micro-FEM model is solved with the aim of the ParFE4 iterative solver. The open source ParFE project provides a multilevel solver for micro-FEM bone structures based on the preconditioning conjugate gradient method.

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−0.2 −0.15 −0.1 −0.05 0 −8 −7 −6 −5 −4 −3 −2 −1 0 ε [microstrains] Stress σ [N/mm 2 ]

Figure 2.11: Stress-Strain curve from experimental failure compressive test. The elasticity apparent modulus is obtained from the slope (in red in the graph) of the initial quasilinear loading region of the curve.

2.5.1 Calculation of Apparent Modulus

Nodal displacements of all verteces of the micro-FEM model as well as the reaction forces at constrained nodes are registered. Considering the scheme shown in figure 2.13, two meaning heights are chosen (z1 and z2 ), these cor-responding to the extensometer rubber bands. For both levels, all nodes with the z coordinate inside the (zlevel± 0,51 voxel size) range are selected. Mean

vertical displacements at levels z1 and z2 are calculated from the displace-ments of the selected nodes. The specimen apparent elastic modulus (EF EM

app

[MPa]) is estimated as follows:

Eapp F EM = R A L2−L1 L1 (2.6) Where R is the total reaction force at the two constrained surfaces and A is the nominal cross section of the bone specimen. L1 and L2 are the distances

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2.5 Solution of FEM model 35

Figure 2.12: Experimental layout scheme for the test of cylindrical cancel-lous bone specimens (diameter = 10 mm; free length = 20 mm). Specimen’s endcaps are embedded into PMMA cement. The two extensometer reference levels are shown. Eventually, a marker system of aluminium spheres allows the repetition of the test along 4 different angolar positions at 0°, 90°, 180°and 270°, respectively, as is shown on the right.

between the extensometer rubber bands before and after solution of the model, respectively.

2.5.2 Back calculation of Tissue Modulus

The effective tissue modulus, in GPa, is finally obtained from the results of FEM analysis and those of mechanical testing, with equation 2.7.

E_tissueEF F ECT IV E = E EXP app EF EM app E_tissueF EM (2.7) Where Eapp exp is the apparent level elastic modulus obtained from mechanical

testing, EF EM

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approxi-Figure 2.13: Specimen scheme: the model is shown before (left) and after (right) its solution. Levels z1 and z2correspond to extensometer rubber bands.

mation model tissue elastic modulus, in GPa.

The procedure presented in this chapter describes how cancellous bone specimens can be characterized using micro-CT scanners. Using the voxels-to-hexahedral elements conversion technique micro-FEM models can be created. These micro-FEM models incorporate the full trabecular architecture and are capable of predicting anisotropic elastic behavior [13].

The advent of powerful computers together with scalable parallel micro-FEM solvers such as ParFE (Copyright (C) 2006 ETH Zurich, [24]) provide a tool for the specimen-specific investigation of the mechanical properties of cancel-lous bone tissue.

The predictive behavior of the ParFE based micro-FEM analysis will be exam-ined in the next chapters as well as its sensitivity to the following parameters:

Reconstruction Voxel Resolution Threshold Value

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Chapter 3 Assessing the ParFE-based

modelling procedure

In the pervious chapter, a procedure for the generation of micro-FEM voxels models of trabecular bone specimens from high resolution micro-CT scanning has been described.

The stack of grey-scale micro-CT attenuation images can be converted to 3D finite element models through the direct generation of equally sized cubic vox-els from image pixvox-els. The result of this operation is a 3D 8-node hexahedral element based FEM model. For a typical cancellous bone specimen (height = 20 mm; diameter = 10 mm) the obtained micro-FEM model (resolution = 39 µm voxel size; threshold grey value = 144) can be described by up to five millions of degrees of freedom.

Even if the discretization of the elasticity equations on this kind of domain is a well-known problem, on the other side the efficient solution of a model characterized by such a remarkable number of nodes is still a demanding issue. The higher computational time and memory required for the solution of a tra-becular bone micro-FEM model makes the use of parallel, distributed memory numerical solvers, a mandatory strategy.

The ParFE Solutor

ParFE is an open source software consisting in a scalable and high paral-37

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lelization finite element solver for bone modeling. It was developed by the Informatics and Biomedical deparments of the ETH of Zurich1_.

To meet the characteristics of most problems, the ParFE software offers two different solution techniques:

Preconditioned Conjiugate Gradient based solutor with assembly of the global stiffness matrix of the model;

Element-by-Element Preconditioned Conjiugate Gradient based solutor. The technique requires equally shaped hexahedral elements: the cubic voxel nature of micro-FEM models fitting properly with this requirement.

The use of a high parallelization FEM software would permit a significant sav-ing of computational time spent for the solution of cancellous bone micro-FEM models. Being the solution of the system of linear equations associated with the model the bottleneck of the whole modelling process, the use of ParFE would produce a significant speed-up of the entire procedure.

3.1 Aim of the study

Previous analyses conduced at IOR’s Medical Technology Laboratory have been supported by the use of Ansysr, a commercial suite for general purpose multiphysics FEM analysis 2_{. A large database of cancellous bone specimens}

had been micro-CT scanned and tested to mechanical compression for the de-termination of apparent cancellous modulus through the procedure described in chapter 2. 3D voxel models had been reconstructed from stacks of micro-CT images.

These micro-FEM models, replicating the geometrical and mechanical charac-teristics of bone specimens as well as the compressive test performed in the laboratory, were solved with Ansys. The so obtained database represented the benchmark for the comparison with analysis performed with ParFE which will be presented in the present study.

Marzio Sala, and all other ParFE developers

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3.2 Materials and Methods 39

On one side Ansys do not offer, at the current version, options for the desirable massive scaling of the process over all the available processors. On the other, the reliability of the ParFE3 _{based modeling procedure, should be assessed on}

an existing cancellous bone database.

In this contest, the aims of the work presented in this chapter were two: 1. To confirm the use, for later studies, of the open source ParFE scalable

software for the modeling of cancellous bone specimens, permitting this way the propitious reduction of total computational time.

This would be obtained by:

finding acceptable agreement with the results previously obtained with the Ansysr Software and

assessing the predictive nature of the results obtained with ParFE for the cancellous tissue bone modulus.

2. To find conformation, at the same time, for the data base of cancel-lous bone tissue modulus previously calculated with an Ansysr based modelling procedure.

3.2 Materials and Methods

3.2.1 Trabecular bone specimens

For this work, 34 cylindrically shaped cancellous bone specimens from human femoral head was used.

The protocol for the specimen’s extraction and its preparation is described in detail in section 2.4 of the previous chapter. No more than a brief summary will be provided here.

Specimens (9.65 to 10.0 mm in diameter and 15.5 to 20.6 mm in free length) were extracted from femur’s diaphyses [27, 28]. A polymethylmethacrylate (PMMA) endcap was applied onto one end of all specimens. Finally, this was placed in Ringers solution for microCT scanning[27].

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3.2.2 Micro-CT scanning

Trabecular specimens were acquired (Skyscan 1072r, BRUKER, Kontich,

Belgium) using a standard protocol [29]: 50kV p, 200µA, 1mm aluminium

filter, exposure time 5.9sec, image averaged on 2 projections, rotation 180 °, rotation step 0.9°, field of view 20mmx20mm and an isotropic pixel size of 19.5µm. Cross-section images were saved in 8 bit format (256 grey levels), 1024 x 1024 pixels in size.

3.2.3 Reconstruction of Micro-FEM models

Hexahedral elements models were reconstructed with a resolution lower than the µ-CT imaging resolution. (Voxel size = 39.0 µm instead of 19.5 µm). Later, in section 4.2, we will discuss the validity of this approximation. A uniform global threshold (grey scale value = 144) and the consequent con-nectivity test were applied to all models. A homogeneous tentative value for tissue elastic modulus was set at 19 GPa.

3.2.4 Experimental tests

The experimental compressive test procedure performed at the Rizzoli’s Ortho-pedic Institute facilities for the determination of the apparent elastic modulus of trabecular bone specimens [27, 28] has been described previously in this document (see section 2.4) and will be only summarized here.

An extensometer was fastened to the central part of the cylinder thanks to two rubber bands (see figure 3.1). Strain rate was maintained at 0.01s−1 [28, 21, 16, 22].

The technique allows the accurate determination of apparent strain: the layout design minimizing end artifacts [15].

3.2.5 Solution of micro-FEM models

Boundary conditions were imposed to the model according with mechanical test layout (see figure 3.1): considering the bottom of the model, displacements of all nodes laying under the PMMA cement height were fully constrained. At

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3.2 Materials and Methods 41

the same time, all nodes laying on the upper surface of the model had hori-zontal displacement forbidden and vertical displacement imposed to simulate the uni-axial compressive behaviour of the experimental test.

FEM models were solved with the use of the ParFe multilevel iterative solver (Iteration limit = 1E6_{; Tolerance = 1E}−11_{). The complete run of a typical}

model (height = 20 mm; diameter = 10 mm; voxel size = 39.0 µm; threshold grey value = 144) required between 20 and 30 minutes and was executed over 32 processors on a dedicated calculus server.

By contrast, the same run with Ansys required at least 4 hours over 4 proces-sors of the same IOR’s server.

Figure 3.1: Experimental layout scheme for compressive test of cylindrical can-cellous bone specimens (diameter = 10 mm; free length = 20 mm). Specimen’s endcaps are embedded into PMMA cement. The two extensometer reference levels are shown. All nodes laying below the PMMA cement are fixed with no degree of freedom. A vertical displacement is imposed to all nodes of the upper surface to reproduce the compressive nature of the experimental test.

Micro-FEM models based on micro-CT reconstructions for the in vitro characterization of the elastic properties of trabecular bone tissue.